Firstly, speak with the Observatory professor, located in the Reception Room of the Observatory (near the Observatory tower). Ask if you can look through the telescope, and he says that it's broken. He tells you that a family of goblins has tampered with the telescope, so ask if you can help. He says that he needs some wood for the tripod, bronze for the tube, and glass for a new lens.

Note: Make sure to talk to the professor three times and give him the molten glass, bronze bar and three planks before attempting to inspect the stove; otherwise, you will have to lure or kill the guard again for the key.

You can find the three planks needed to fix the telescope just north of the Barbarian Outpost, which is to the far north of Ardougne, or in Port Khazard, if you don't already have them with you. A bronze bar can be smithed by anyone, or you can buy one. A piece of molten glass can be made by using your Craftingskill. Once you have these three items, bring them back to the professor who will take them from you. He then says that he needs a mould so that he can make a new lens.

If you don't have Antipoison potions, you can pick up one Superantipoison at the spawn in the southeastern building. Now, go down the stairs to the east of the Reception Room to find the Observatory Dungeon (the room with the stairs is directly east of the start point).

Once down the stairs, you must find the goblin kitchen key, which is commonly found in the south-eastern most chest, although it can vary from person to person. Search any chests you pass on the way to the south-eastern chest just in case. Keep searching the chests (refer to the map) until you find the key. Sometimes, there may be a poisonous spider inside the chest in which case run away quickly or you may get poisoned. There is also a chest north-east of the stairs to the Observatory, which spawns infinite Antipoison(1) vials, which you can combine to make full vials. Consider using this moment to create yourself a stock for future use.

Now that you've got the key, head all the way back to the kitchen gate that you may have passed (take the northeastern path from entrance), and "prod" the sleeping guard. He will wake up and attack you. You can either defeat him or get him stuck behind a wall.

Note: Getting him stuck behind a wall will allow you to go through the door. However, if you pass the goblin again, you will be attacked. Also, another goblin will spawn asleep. You can fight multiple goblins.

Open the kitchen gate (you will use the key and discard it), and then inspect the goblin stove. You determine the goblins are cooking something foul in the lens mould. Dump out the stew and bring this back to the professor, who now says that he doesn't have a high enough Crafting level to make the lens for the telescope. He gives you the molten glass and the mould back, so just use the molten glass with the lens mould to craft him a new lens. Give it back to him, and he wants you to visit him in the Observatory.

Climb back down the stairs into the maze. Now, instead of taking the south-east path, follow the north-west passage (past the kitchen) to find some stairs leading to the Observatory. A cutscene will play. After the cutscene ends, go up the stairs and talk to the professor, and then "view" the telescope to see a constellation.

Looking through the telescope.

After closing the window, the professor should ask you which constellation you saw. The constellations are on wall charts in the main level of the Observatory. Identify your constellation, talk to the professor again, and you should get your reward in your backpack!